Hi Pankaj, it's hard to give a short answer to this question. The main difference between the meshfree and FEM is the shape functions used to approximate the trial and test functions of discretization. The key is using overlapping domains in meshfree methods, which gives more support nodes for each point, allowing a richer approximation and avoiding any artificial discontinuity in the field.

In continuum mechanics, where FEM is widely used, the most similar comparison can be made when using the Bubnov-Galerkin discretization of the continuum equations. In this case, the integrals can be computed with a similar Gauss quadrature scheme and the precision of both methods can be directly compared. Take in mind that the meshfree shape functions are not defined in an isoparametric space, which means that a linear system must be solved for each shape function. If you use an Updated Lagrangian formulation, the shape functions would need an update in every step of the nonlinear solver procedure. It is the main reason why you´ll find the Total Lagrangian as a preferred form when dealing with meshfree approximations.

Of course all of this has lots of implications in the internals of the programming and also in the numerical loops. In summary, there are some situations when the increased precision and smoothness makes up for the extra computational effort. Also, meshfree methods give a lot more ways of approaching the problem to solve, as the interpolation nodes and integration points can be completely uncoupled in space.

If you want to read further, take a look to some applications on multibody systems simulations using nonlinear mechanical and thermal formulations, respectively, in the following links:

https://www.researchgate.net/publication/260136371_A_meshfree_application_to_the_nonlinear_dynamics_of_flexible_multibody_systems?ev=prf_pub

https://www.researchgate.net/publication/260821585_Application_of_Galerkin_meshfree_methods_to_nonlinear_thermo-mechanical_simulation_of_solids_under_extremely_high_pulsed_loading?ev=prf_pub

Best regards,

Daniel

Conference Paper A meshfree application to the nonlinear dynamics of flexible...

Article Application of Galerkin meshfree methods to nonlinear thermo...

Hi,

FEM vs. FDM vs meshless DAM and MLPG comparison is done in

https://www.researchgate.net/publication/257396907_Comparison_of_local_weak_and_strong_form_meshless_methods_for_2-D_diffusion_equation

The accuracy of strong form meshless method is exactly the same as FDM and if you code it right the execution time will be also similar. Still there are several advantages in using meshless methods. You can easily increase the support domain, change basis, adopt nodal densities, etc. These features can be quite useful when you are dealing with anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations

However, the strong form meshless methods such as LRBFCM and DAM can be pretty much understood as a generalization of FDM. If you use small support domains, monomials for basis and collocation the LRBFCM method is practically FDM. I can send you some codes and examples if you want. You will find some more papers related to meshless methods in my publications list ... I can also send you some other papers.

Perhaps, if you tell me what you would like to solve we could discuss more.

best regards

Grega

Article Comparison of local weak and strong form meshless methods fo...

Yes of course, [email protected]

I disagree that meshfre radial basis functions take more time. Some meshfree metodslook like finite elements-this is not true. If you want to dispite this, contact me at [email protected]

@Edward -  When I had asked this question, I was learning element free Galerkin's method which actually are slower than FDM. However, over the years, I have seen papers showing strong form meshless collocation faster than FEM and FDM. I have edited the question details.

Thank you for replying.

Readers may read the following papers:

C.-S. Huang, C.-F. Lee, A.H.-D. Cheng: Eng. Anal. Bound. Elem:

31:614.623, (2007).

C.-S. Huang, H.-D. Yen, A.H.-D. Cheng: Eng. Anal. Bound. Elem.,

34:802-809, 2010.

Because C^inf radial basis functions are exponentially convergent,  the standard methods are horribly slow. What counts is the TOTAL effort to complete the task, not the unit time/operation.

Secondly, some meshfree methods are basically finite elements presented differently,

still using low order polynomial basis functions.

Its a tricky question to answer.

Firstly , this method is applicable/useful when we need to have mesh independence or where problems are constrained due to mesh. For Example

• Fracture Problems ( Most Popular) 
• Problems where we have interfaces ( Shell with stiffeners)
• Joints
• Large deformation problems
• Problems where we have steep gradients in the material across thickness (FGM)

These are the places where higher order approximations can come handy. This is where Meshless methods score over FGM/FDM. We can assume higher order approximations and itterate the domain of influence to get the desired results.

But , all meshless methods are not suitable for every kind of problems. We need to have a background mesh for integration. Numerical integration at times becomes cumbersome and time consuming.

Its a tricky question to answer.

Firstly , this method is applicable/useful when we need to have mesh independence or where problems are constrained due to mesh. For Example

• Fracture Problems ( Most Popular) 
• Problems where we have interfaces ( Shell with stiffeners)
• Joints
• Large deformation problems
• Problems where we have steep gradients in the material across thickness (FGM)

These are the places where higher order approximations can come handy. This is where Meshless methods score over FGM/FDM. We can assume higher order approximations and itterate the domain of influence to get the desired results.

But , all meshless methods are not suitable for every kind of problems. We need to have a background mesh for integration. Numerical integration at times becomes cumbersome and time consuming.

One of the earliest meshfree method is smoothed particle hydrodynamics which was originally invented for modeling astrophysical phenomena in 1977. Grid-based method faces difficulties when we deal with explosion, especially in space research. Also, it is difficult to handle the high velocity impact due to large deformations and free surfaces. For more detailed discussion about comparison, I recommend to read the first chapter of the following book.

Smoothed Particle Hydrodynamics:A meshfree particle method

https://books.google.ie/books?id=pfcWnwEACAAJ&dq=isbn:9812384561&hl=en&sa=X&ved=0ahUKEwi6lb7Zme7NAhXsAMAKHRSrAMQQ6AEIHTAA